Redundancy in Logic I: CNF Propositional Formulae

A knowledge base is redundant if it contains parts that can be inferred from the rest of it. We study the problem of checking whether a CNF formula (a set of clauses) is redundant, that is, it contains clauses that can be derived from the other ones. Any CNF formula can be made irredundant by deleting some of its clauses: what results is an irredundant equivalent subset (I.E.S.) We study the complexity of some related problems: verification, checking existence of a I.E.S. with a given size, checking necessary and possible presence of clauses in I.E.S.'s, and uniqueness. We also consider the problem of redundancy with different definitions of equivalence.Comment: Extended and revised version of a paper that has been presented at ECAI 200

Beyond Well-designed SPARQL

SPARQL is the standard query language for RDF data. The distinctive feature of SPARQL is the OPTIONAL operator, which allows for partial answers when complete answers are not available due to lack of information. However, optional matching is computationally expensive - query answering is PSPACE-complete. The well-designed fragment of SPARQL achieves much better computational properties by restricting the use of optional matching - query answering becomes coNP-complete. However, well-designed SPARQL captures far from all real-life queries - in fact, only about half of the queries over DBpedia that use OPTIONAL are well-designed. In the present paper, we study queries outside of well-designed SPARQL. We introduce the class of weakly well-designed queries that subsumes well-designed queries and includes most common meaningful non-well-designed queries: our analysis shows that the new fragment captures about 99% of DBpedia queries with OPTIONAL. At the same time, query answering for weakly well-designed SPARQL remains coNP-complete, and our fragment is in a certain sense maximal for this complexity. We show that the fragment\u27s expressive power is strictly in-between well-designed and full SPARQL. Finally, we provide an intuitive normal form for weakly well-designed queries and study the complexity of containment and equivalence

Computational complexity of counting coincidences

Can you decide if there is a coincidence in the numbers counting two different combinatorial objects? For example, can you decide if two regions in $\mathbb{R}^3$ have the same number of domino tilings? There are two versions of the problem, with $2\times 1 \times 1$ and $2\times 2 \times 1$ boxes. We prove that in both cases the coincidence problem is not in the polynomial hierarchy unless the polynomial hierarchy collapses to a finite level. While the conclusions are the same, the proofs are notably different and generalize in different directions. We proceed to explore the coincidence problem for counting independent sets and matchings in graphs, matroid bases, order ideals and linear extensions in posets, permutation patterns, and the Kronecker coefficients. We also make a number of conjectures for counting other combinatorial objects such as plane triangulations, contingency tables, standard Young tableaux, reduced factorizations and the Littlewood--Richardson coefficients.Comment: 23 pages, 6 figure

Reasoning about distributed relational data and query evaluation

Large data sets are often stored distributedly to increase the reliability of systems and the efficiency of query evaluation in them. While some query operators -- like selections and projections -- are intrinsically conform with parallel evaluation, others -- like joins -- demand specific distribution patterns. For relational databases, a common approach to evaluate queries in parallel relies on the use of rather simple distribution patterns for binary joins and the computation of the query result according to some query plan, operator by operator. Often, this requires the redistribution of large intermediate results (possibly larger than the input and/or output) and thus may lead to unnecessary long processing times. Thus, especially in the last decade, more elaborate distribution patterns that depend on the whole query have been studied and shown to allow more efficient query evaluation in several cases by reducing the amount of communication between servers. Ameloot et al. have described a setting where query evaluation is studied for a broad range of distribution patterns. Their work focuses on problems to decide whether a query can be evaluated correctly under a given distribution pattern. More particularly, they have considered two problems: "parallel correctness", where the pattern is specified explicitly, and "parallel-correctness transfer", where the pattern is known to be appropriate for another query. This thesis comprises the author's contributions to the complexity-theoretical investigation of these problems for conjunctive queries (and extensions thereof). These contributions complement the main characterisations and some additional complexity results by Ameloot et al. Furthermore, this thesis contains some new characterisations for "polarised" queries. Via the characterisations, parallel correctness and parallel-correctness transfer can be translated into questions on the co-occurrences of certain facts, induced by the query, on some server. Such questions and others can be modelled by "distribution dependencies", a variant of the well-known tuple- and equality-generating dependencies. Modelling via these constraints allows a more general description of distribution patterns in distributed relational data. The third contribution of this thesis is the study of the implication problem for distribution dependencies, providing lower and upper bounds for some fragments

Equilibria, Fixed Points, and Complexity Classes

Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in polynomial time. There are certain common computational principles underlying different types of equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP. Representative complete problems for these classes are respectively, pure Nash equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria in 2-player normal form games, and (mixed) Nash equilibria in normal form games with 3 (or more) players. This paper reviews the underlying computational principles and the corresponding classes